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June 22

Generalization of a Certain Formula for Pi

Let $a_0=0$ and $a_{n+1}=\sqrt{A+B{a}_n}.$ Then, assuming convergence, we have $a_{\mathrm{\infty }}=\ell =\frac{B+\sqrt{B^2+4A}}{2}.$ Thus, for $A=B=1$ we have $\ell =\varphi ,$ for instance. Now, for $A=B=\frac{1}{2}$ we have $\ell =1,$ and $\underset{n\to \mathrm{\infty }}{\lim }{2}^n\sqrt{\frac{\ell -a_n}{2}}=\frac{\pi }{4}.$ My question would be with what constant to replace $2$ in general, for different values of A and B, so that the limit in question should converge to a finite non-zero quantity. In other words, if $f(\frac{1}{2},\frac{1}{2})=2,$ what is the general formula for $f(A,B)$ ? Thank you. — 79.118.171.25 (talk) 22:57, 22 June 2015 (UTC)

Apparently, $f(1,1)=\sqrt{1+\sqrt{5}},$ and the limit in question is the square root of the Paris constant. — 79.118.171.25 (talk) 03:07, 23 June 2015 (UTC)

I get $f(A,B)=\sqrt{2\ell /B}$. My idea is to let $x_n=\ell -a_n$ and write its recurrence formula. The behavior for small $x$ is dictated by the linear term of its Maclaurin series. $\ell -\sqrt{A+B(\ell -x)}=0+\frac{B}{2\ell }x+...$ From there it's not hard to understand the behavior of $\sqrt{\frac{x_n}{2}}$ and find $f(A,B)$. Egnau (talk) 03:40, 23 June 2015 (UTC)

I arrived just these past few minutes at the same conclusion, and wanted to post it, but was unable to connect. :-) Thanks ! — 79.113.226.120 (talk) 03:53, 23 June 2015 (UTC)

And in general, for $a_{n+1}=\sqrt[m]{A+B{a}_n}$ we have $f_m(A,B)=\sqrt{\frac{m{\lambda }^{m-1}}{B}},$ where $\lambda$ is a root of $t^m=A+Bt.$ — 79.113.226.120 (talk) 10:08, 23 June 2015 (UTC)